| L(s) = 1 | + (−2.60 + 0.846i)2-s + (−1.86 − 2.56i)3-s + (4.45 − 3.23i)4-s + (0.0540 + 2.23i)5-s + (7.02 + 5.10i)6-s − 2.47i·7-s + (−5.65 + 7.77i)8-s + (−2.17 + 6.69i)9-s + (−2.03 − 5.77i)10-s + (−1.18 − 3.65i)11-s + (−16.6 − 5.39i)12-s + (−1.34 − 0.437i)13-s + (2.09 + 6.45i)14-s + (5.63 − 4.30i)15-s + (4.73 − 14.5i)16-s + (0.587 − 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−1.84 + 0.598i)2-s + (−1.07 − 1.48i)3-s + (2.22 − 1.61i)4-s + (0.0241 + 0.999i)5-s + (2.86 + 2.08i)6-s − 0.936i·7-s + (−1.99 + 2.74i)8-s + (−0.725 + 2.23i)9-s + (−0.643 − 1.82i)10-s + (−0.357 − 1.10i)11-s + (−4.79 − 1.55i)12-s + (−0.373 − 0.121i)13-s + (0.560 + 1.72i)14-s + (1.45 − 1.11i)15-s + (1.18 − 3.64i)16-s + (0.142 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0297489 + 0.0461334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0297489 + 0.0461334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.0540 - 2.23i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| good | 2 | \( 1 + (2.60 - 0.846i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.86 + 2.56i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 2.47iT - 7T^{2} \) |
| 11 | \( 1 + (1.18 + 3.65i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.34 + 0.437i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (0.679 + 0.494i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (1.51 - 0.491i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.61 - 3.35i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.88 + 2.82i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.34 - 1.41i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (0.386 - 1.18i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.80iT - 43T^{2} \) |
| 47 | \( 1 + (-2.80 - 3.86i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.10 + 1.51i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.27 - 3.92i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.05 - 9.39i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (9.16 - 12.6i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.0639 + 0.0464i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.40 + 1.10i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.953 + 0.692i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (5.68 - 7.82i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.37 - 4.24i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.481 - 0.662i)T + (-29.9 + 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06182362741732949946358927694, −10.75326304319906461630430977186, −9.774045466752906894583258526803, −8.338509723851337857331649301525, −7.45106785378935807801194598103, −7.22671662177043567468605315154, −6.22024549601932258561217717020, −5.66523928587387560348653233165, −2.59779347032806116233488038823, −1.18981113003678581011568861592,
0.079010798850388007122931474829, 2.07903033387617666073997051285, 3.82656474514832448678283922770, 5.10128092814151693334958603977, 6.13372996915406176229762563120, 7.55024620057058015005312659468, 8.730307275803253698034000721838, 9.323107840644781555628677379588, 9.885097278314181588460819036398, 10.60684295954781884902187332880
